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A270599 Number of ways to express 1 as the sum of unit fractions with odd denominators such that the sum of those denominators is n. 2
1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,33

COMMENTS

Number of partitions of n into such odd parts that the sum of their reciprocals is one. - Antti Karttunen, Jul 23 2018

It would be nice to know whether nonzero values may occur only on n of the form 8k+1.

LINKS

David A. Corneth, Table of n, a(n) for n = 1..370 (terms 1..150 from Seiichi Manyama, terms 150..273 from Antti Karttunen)

David A. Corneth, Tuples up to n = 370.

Index entries for sequences related to Egyptian fractions

FORMULA

a(2*k) = 0. - David A. Corneth, Jul 24 2018

EXAMPLE

1 = 1/3 + 1/3 + 1/3, the sum of denominators is 9, this is the only expression of 1 as unit fractions with odd denominators that sum to 9, so a(9)=1.

1 = 1/15 + 1/5 + 1/5 + 1/5 + 1/3 = 1/9 + 1/9 + 1/9 + 1/3 + 1/3 are the only solutions with odd denominators that sum to 33, thus a(33) = 2. - Antti Karttunen, Jul 24 2018

MATHEMATICA

Array[Count[IntegerPartitions[#, All, Range[1, #, 2]], _?(Total[1/#] == 1 &)] &, 70] (* Michael De Vlieger, Jul 26 2018 *)

PROG

(Ruby)

def f(n)

  n - 1 + n % 2

end

def partition(n, min, max)

  return [[]] if n == 0

  [f(max), f(n)].min.step(min, -2).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}

end

def A270599(n)

  ary = [1]

  (2..n).each{|m|

    cnt = 0

    partition(m, 2, m).each{|ary|

      cnt += 1 if ary.inject(0){|s, i| s + 1 / i.to_r} == 1

    }

    ary << cnt

  }

  ary

end

(PARI) A270599(n, maxfrom=n, fracsum=0) = if(!n, (1==fracsum), my(s=0, tfs, k=(maxfrom-!(maxfrom%2))); while(k >= 1, tfs = fracsum + (1/k); if(tfs > 1, return(s), s += A270599(n-k, min(k, n-k), tfs)); k -= 2); (s)); \\ Antti Karttunen, Jul 23 2018

(PARI)

\\ More verbose version for computing values of a(n) for large n:

A270599(n) = if(!(n%2), 0, my(s=0); forstep(k = n, 1, -2, print("A270599(", n, ") at toplevel, k=", k, " s=", s); s += A270599aux(n-k, min(k, n-k), 1/k)); (s));

A270599aux(n, maxfrom, fracsum) = if(!n, (1==fracsum), my(s=0, tfs, k=(maxfrom-!(maxfrom%2))); while(k >= 1, tfs = fracsum + (1/k); if(tfs > 1, return(s), s += A270599aux(n-k, min(k, n-k), tfs)); k -= 2); (s)); \\ Antti Karttunen, Jul 24 2018

CROSSREFS

Cf. A000009, A051908.

Cf. also A201644, A201646, A201647, A201648, A201649.

Sequence in context: A331984 A087781 A181009 * A091398 A062103 A112314

Adjacent sequences:  A270596 A270597 A270598 * A270600 A270601 A270602

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Mar 26 2016

EXTENSIONS

Name corrected by Antti Karttunen, Jul 23 2018 at the suggestion of David A. Corneth

STATUS

approved

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Last modified April 18 19:31 EDT 2021. Contains 343089 sequences. (Running on oeis4.)